Dualities and isomorphisms of classical groups
Four of the five families of classical generalised quadrangles come in dual pairs: (i) and ; (ii) and . Both can be demonstrated by the Klein correspondence. Recall from the last post that the Klein correspondence maps a line of represented as the row space of
to the point where
Now consider the symplectic generalised quadrangle defined by the bilinear alternating form
A totally isotropic line must then satisfy
Therefore, the lines of are mapped to points of lying in the hyperplane . Now the quadratic form defining is and is the tangent hyperplane at the projective point , which does not lie in the quadric. Hence the hyperplane is non-degenerate and so we see that maps to points of . That this mapping is bijective follows from noting that the number of lines of is equal to the number of points of (namely, ).
Now we will consider a more difficult situation which reveals that the generalised quadrangles and are also formally dual to one another. We could take the approach above, but instead, we will use a cunning way (told to me by Tim Penttila) to produce an Hermitian form on . First we consider as the subspaces of that are fixed by a particular involution known as a Baer involution. In order for this to look nice, I will change the form that defines :
where is an irreducible homogeneous quadratic in two variables over . So when we restrict this form to the points with entries only in , we have a form of minus type; a polar space isomorphic to . The Baer involution is simply the map which raises each coordinate to its q-th power, and so our is exactly the fixed singular elements of this involution. Moreover, switches the latins and greeks. To see this, note that the latins and greeks form a block-system for the full orthogonal group. Now consider the following singular plane:
Then applying to each point comprising will give us a different plane which intersects in the line
Therefore, is in a different class to . Furthermore, if we take a line of , it is fixed by and the two singular planes incident with this line are interchanged by . If we relate what is doing back to , we see that it is inducing a polarity of having absolute lines. By the classification of polarities, this is a unitary polarity. So we see that corresponds to the hermitian surface . Hence .
Ovoids and spreads
An ovoid of is a set of points such that no two are collinear in a line of . So under the Klein correspondence, we have a set of lines of such that no two are concurrent in a point of (and equivalently, they do not span a plane of ). So an ovoid of is equivalent to a spread of .
We also mention here a beautiful observation of Shult and Thas:
any 1-system of , q odd, is equivalent to the classical 1-system (arising from the Klein image of an elliptic quadric).
A 1-system of is a set of lines of that are pairwise opposite: there is no singular plane on one of the lines meeting the other in a point. Under the Klein correspondence, we end up with a set of point/plane incident pairs, such that no two will intersect in a line that passes through their given points. Such a configuration defines an ovoid of : a set of points with no 3 collinear. By a result attributed to the independent work of Barlotti and Panella (1955), this ovoid is projectively equivalent to en elliptic quadric. So the 1-system is the Klein image of an elliptic quadric.
Isomorphisms of simple gorups
The Klein Correspondence yields a simple way to explain why and . First, consider an eight dimensional vector space over equipped with the quadratic form
Think of the vectors as eight-tuples, so that we have a natural action of the symmetric group on the vectors: simply by permutation of coordinates. Then fixes and the subspace of vectors whose coordinates sum to . Then induces an action on the quotient six-dimensional vector space
We should also observe that preserves the quadratic form , and the inherited quadratic form on . So has a representation into , and then by orders, we have an isomorphism. The Klein correspondence then yields .
Now for , we observe that we can use the same setting since can be realised inside the stabiliser of a point or a plane of . Take the following singular plane of :
Now does not fix this plane by acting on coordinates, but we do have a subgroup of , namely that acts 2-transitively on these coordinates if we think of its natural action on the projective line labelled in the order . Then by comparing orders, we see that is isomorphic to .
See also Rob Wilson’s book “The Finite Simple Groups”, page 99.